In 1936, Witt gave a criterion for embedding a V4-extension into a
Q8-extension in terms of an equivalence of quadratic forms. Using
this result, it is possible to give similar criteria for solving other
embedding problems, starting with embedding a D4-extension into a
QD8-extension, where D4 and QD8 are the dihedral and
quasi-dihedral groups of order 8 and 16, resp. This again permits
a
complete description of extensions with certain 2-groups of
order 16 as Galois groups.